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Published byCecil Floyd Modified over 4 years ago

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Chapter 1: Preliminary Information Section 1-1: Sets of Numbers

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Objectives Given the name of a set of numbers, provide an example. Given an example, name the sets to which the number belongs.

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Two main sets of numbers Real Numbers ◦ Used for “real things” such as: Measuring Counting ◦ Real numbers are those that can be plotted on a number line Imaginary Numbers- square roots of negative numbers

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The Real Numbers Rational Numbers-can be expressed exactly as a ratio of two integers. This includes fractions, terminating and repeating decimals. ◦ Integers- whole numbers and their opposites ◦ Natural Numbers- positive integers/counting numbers ◦ Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Irrational Numbers-Irrational numbers are those that cannot be expressed exactly as a ratio of two numbers ◦ Square roots, cube roots, etc. of integers ◦ Transcendental numbers-numbers that cannot be expressed as roots of integers

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Chapter 1: Preliminary Information Section 1-2: The Field Axioms

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Objective Given the name of an axiom that applies to addition or multiplication that shows you understand the meaning of the axiom.

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The Field Axioms Closure Commutative Property Associative Property Distributive Property Identity Elements Inverses

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Closure {Real Numbers} is closed under addition and under multiplication. That is, if x and y are real numbers then: ◦ x + y is a unique real number ◦ xy is a unique real number

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More on Closure Closure under addition means that when two numbers are chosen from a set, the sum of those two numbers is also part of that same set of numbers. For example, consider the digits. ◦ The digits include 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. ◦ If the digits are closed under addition, it means you can pick any two digits and their sum is also a digit. ◦ Consider 8 + 9 The sum is 17 Since 17 is not part of the digits, the digits are not closed under addition.

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More on Closure Closure under multiplication means that when two numbers are chosen from a set, the product of those two numbers is also part of that same set of numbers. For example, consider the negative numbers. ◦ If we choose -6 and -4 we multiply them and get 24. ◦ Since 24 is not a negative number, the negative numbers are not closed under multiplication.

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The Commutative Property Addition and Multiplication of real numbers are commutative operations. That means: ◦ x + y = y + x ◦ xy =yx Are subtraction and division commutative?

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Associative Property Addition and Multiplication of real numbers are associative operations. That means: ◦ (x + y) + z = x + (y + z) ◦ (xy)z = x(yz)

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Distributive Property Multiplication distributes over addition. That is, if x, y and z are real numbers, then: x (y + z) = xy + xz Multiplication does not distribute over multiplication!

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Identity Elements The real numbers contain unique identity elements. ◦ For addition, the identity element is 0. ◦ For multiplication, the identity element is 1.

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Inverses The real numbers contain unique inverses ◦ The additive inverse of any number x is the number – x. ◦ The multiplicative inverse of any number x is 1/x, provided that x is not 0.

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